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Thursday, August 30, 2018

**Abstract:** Ramanujan was fascinated by equal sums of two cubes, as shown in the legendary anecdote about $1729 = 10^3 + 9^3 = 12^3 + 1^3$. He was also interested in the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$ for polynomials $f_j$ and gave several examples in which the $f_j$'s were quadratic polynomials. We will discuss the complete solution for quadratic complex polynomials, with side trips into elliptic curves and computational algebraic geometry. The simplest such identity is $(x^2 + x y - y^2)^3 + (x^2 - x y - y^2)^3 = 2x^6 -2y^6$. This talk is meant to be accessible to first-year graduate students.